Submanifolds of Euclidean space with parallel mean curvature vector
The object of the paper is to study some compact submanifolds in the Euclidean space Rn whose mean curvature vector is parallel in the normal bundle. First we prove that there does not exist an n-dimensional compact simply connected totally real submanifold in R2n whose mean curvature vector is para...
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| Format: | Article |
| Language: | English |
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Wiley
1991-01-01
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| Series: | International Journal of Mathematics and Mathematical Sciences |
| Subjects: | |
| Online Access: | http://dx.doi.org/10.1155/S0161171291000728 |
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| _version_ | 1832548093199908864 |
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| author | Tahsin Ghazal Sharief Deshmukh |
| author_facet | Tahsin Ghazal Sharief Deshmukh |
| author_sort | Tahsin Ghazal |
| collection | DOAJ |
| description | The object of the paper is to study some compact
submanifolds in the Euclidean space Rn whose mean curvature
vector is parallel in the normal bundle. First we prove that
there does not exist an n-dimensional compact simply connected
totally real submanifold in R2n whose mean curvature vector is
parallel. Then we show that the n-dimensional compact totally
real submanifolds of constant curvature and parallel mean
curvature in R2n are flat. Finally we show that compact
Positively curved submanifolds in Rn with parallel mean
curvature vector are homology spheres. The last result in
particular for even dimensional submanifolds implies that their
Euler poincaré characteristic class is positive, which for the
class of compact positively curved submanifolds admiting isometric
immersion with parallel mean curvature vector in Rn, answers the
problem of Chern and Hopf |
| format | Article |
| id | doaj-art-e51001aa8e2a48b9ada7379d6e32752f |
| institution | Kabale University |
| issn | 0161-1712 1687-0425 |
| language | English |
| publishDate | 1991-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | International Journal of Mathematics and Mathematical Sciences |
| spelling | doaj-art-e51001aa8e2a48b9ada7379d6e32752f2025-02-03T06:42:23ZengWileyInternational Journal of Mathematics and Mathematical Sciences0161-17121687-04251991-01-0114353353610.1155/S0161171291000728Submanifolds of Euclidean space with parallel mean curvature vectorTahsin Ghazal0Sharief Deshmukh1Department of Mathematics, College of Science, P.O. Box 2455, King Saud University, Riyadh 11451, Saudi ArabiaDepartment of Mathematics, College of Science, P.O. Box 2455, King Saud University, Riyadh 11451, Saudi ArabiaThe object of the paper is to study some compact submanifolds in the Euclidean space Rn whose mean curvature vector is parallel in the normal bundle. First we prove that there does not exist an n-dimensional compact simply connected totally real submanifold in R2n whose mean curvature vector is parallel. Then we show that the n-dimensional compact totally real submanifolds of constant curvature and parallel mean curvature in R2n are flat. Finally we show that compact Positively curved submanifolds in Rn with parallel mean curvature vector are homology spheres. The last result in particular for even dimensional submanifolds implies that their Euler poincaré characteristic class is positive, which for the class of compact positively curved submanifolds admiting isometric immersion with parallel mean curvature vector in Rn, answers the problem of Chern and Hopfhttp://dx.doi.org/10.1155/S0161171291000728submanifoldstotally real submanifolds homology sphereEuler-poincaré characteristic. |
| spellingShingle | Tahsin Ghazal Sharief Deshmukh Submanifolds of Euclidean space with parallel mean curvature vector International Journal of Mathematics and Mathematical Sciences submanifolds totally real submanifolds homology sphere Euler-poincaré characteristic. |
| title | Submanifolds of Euclidean space with parallel mean curvature vector |
| title_full | Submanifolds of Euclidean space with parallel mean curvature vector |
| title_fullStr | Submanifolds of Euclidean space with parallel mean curvature vector |
| title_full_unstemmed | Submanifolds of Euclidean space with parallel mean curvature vector |
| title_short | Submanifolds of Euclidean space with parallel mean curvature vector |
| title_sort | submanifolds of euclidean space with parallel mean curvature vector |
| topic | submanifolds totally real submanifolds homology sphere Euler-poincaré characteristic. |
| url | http://dx.doi.org/10.1155/S0161171291000728 |
| work_keys_str_mv | AT tahsinghazal submanifoldsofeuclideanspacewithparallelmeancurvaturevector AT shariefdeshmukh submanifoldsofeuclideanspacewithparallelmeancurvaturevector |